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Understanding tensor products and $R$-algebras

Sorry if this question isn't quite precise. Anyways, I'm reviewing for my algebra final right now and two things I don't think I quite grasp as well as I'd like are tensor products and $R$-algebras. I ...
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Is it to the students' advantage to learn the language of infinitesimals?

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...